This book contains the papers developing out the presentations given at the International Conference organized by the Torino Academy of Sciences and the Department of Mathematics Giuseppe Peano of the Torino University to celebrate the 150th anniversary of G. Peano's birth - one of the greatest figures in modern mathematics and logic and the most important mathematical logician in Italy - a century after the publication of Formulario Mathematico, a great attempt to systematise Mathematics in symbolic form.

Many literary critics seem to think that an hypothesis about obscure and remote questions of history can be refuted by a simple demand for the production of more evidence than in fact exists. The demand is as easy to make as it is impossible to satisfy. But the true test of an hypothesis, if it cannot be shown to con?ict with known truths, is the number of facts that it correlates and explains. Francis M. Cornford [1914] 1934, 220. It was in the autumn of 1997 that the research project leading to this publication began. One of us [GH], while a visiting fellow at the Center for Philosophy of Science (University of Pittsburgh), gave a talk entitled, "Proportions and Identity: The Aesthetic Aspect of Symmetry". The presentation focused on a confusion s- rounding the concept of symmetry: it exhibits unity, yet it is often claimed to reveal a form of beauty, namely, harmony, which requires a variety of elements. In the audience was the co-author of this book [BRG] who responded with enthusiasm, seeking to extend the discussion of this issue to historical sources in earlier periods. A preliminary search of the literature persuaded us that the history of symmetry was rich in possibilities for new insights into the making of concepts. John Roche's brief essay (1987), in which he sketched the broad outlines of the history of this concept, was particularly helpful, and led us to conclude that the subject was worthy of monographic treatment.

Intellectual History and the Identity of John Dee In April 1995, at Birkbeck College, University of London, an interdisciplinary colloquium was held so that scholars from diverse fields and areas of expertise could 1 exchange views on the life and work of John Dee. Working in a variety of fields - intellectual history, history of navigation, history of medicine, history of science, history of mathematics, bibliography and manuscript studies - we had all been drawn to Dee by particular aspects of his work, and participating in the colloquium was to c- front other narratives about Dee's career: an experience which was both bewildering and instructive. Perhaps more than any other intellectual figure of the English Renaissance Dee has been fragmented and dispersed across numerous disciplines, and the various attempts to re-integrate his multiplied image by reference to a particular world-view or philosophical outlook have failed to bring him into focus. This volume records the diversity of scholarly approaches to John Dee which have emerged since the synthetic accounts of I. R. F. Calder, Frances Yates and Peter French. If these approaches have not succeeded in resolving the problematic multiplicity of Dee's activities, they will at least deepen our understanding of specific and local areas of his intellectual life, and render them more historiographically legible.

An elegantly dramatized and illustrated dialog on the square root of two and the whole concept of irrational numbers.

Originally published in 1987, this important synthesis represented the first effort by modern scholars to convey the variety of ways in which medieval scientists and natural philosophers used mathematics and mathematical modes of thought to describe natural phenomena. Eleven distinguished historians of science contributed original essays on the application of mathematics to natural philosophy, astronomy, cosmology, optics and medicine. The book is a fitting tribute to Professor Marshall Clagett of The Institute for Advanced Study, Princeton, for his significant contributions to the history of medieval science.

Through the prism of gender, this text explores the contrasting cultures and practice of mathematics and science and asks how they impacted on women. Claire Jones assesses nineteenth-century ideas about women's intellect, femininity and masculinity, and assesses how these attitudes shaped women's experiences as students and practitioners.

Trigonometry has always been an underappreciated branch of mathematics. It has a reputation as a dry and difficult subject, a glorified form of geometry complicated by tedious computation. In this book, Eli Maor draws on his remarkable talents as a guide to the world of numbers to dispel that view. Rejecting the usual arid descriptions of sine, cosine, and their trigonometric relatives, he brings the subject to life in a compelling blend of history, biography, and mathematics. He presents both a survey of the main elements of trigonometry and a unique account of its vital contribution to science and social development. Woven together in a tapestry of entertaining stories, scientific curiosities, and educational insights, the book more than lives up to the title "Trigonometric Delights."

Maor, whose previous books have demystified the concept of infinity and the unusual number "e," begins by examining the "proto-trigonometry" of the Egyptian pyramid builders. He shows how Greek astronomers developed the first true trigonometry. He traces the slow emergence of modern, analytical trigonometry, recounting its colorful origins in Renaissance Europe's quest for more accurate artillery, more precise clocks, and more pleasing musical instruments. Along the way, we see trigonometry at work in, for example, the struggle of the famous mapmaker Gerardus Mercator to represent the curved earth on a flat sheet of paper; we see how M. C. Escher used geometric progressions in his art; and we learn how the toy Spirograph uses epicycles and hypocycles.

Maor also sketches the lives of some of the intriguing figures who have shaped four thousand years of trigonometric history. We meet, for instance, the Renaissance scholar Regiomontanus, who is rumored to have been poisoned for insulting a colleague, and Maria Agnesi, an eighteenth-century Italian genius who gave up mathematics to work with the poor--but not before she investigated a special curve that, due to mistranslation, bears the unfortunate name "the witch of Agnesi." The book is richly illustrated, including rare prints from the author's own collection. "Trigonometric Delights" will change forever our view of a once dreaded subject.

Since its publication, C.F. Gauss's Disquisitiones Arithmeticae (1801) has acquired an almost mythical reputation, standing as an ideal of exposition in notation, problems and methods; as a model of organisation and theory building; and as a source of mathematical inspiration. Eighteen authors - mathematicians, historians, philosophers - have collaborated in this volume to assess the impact of the Disquisitiones, in the two centuries since its publication.

The 1947 paper by John von Neumann and Herman Goldstine, "Numerical Inverting of Matrices of High Order" (Bulletin of the AMS, Nov. 1947), is considered as the birth certificate of numerical analysis. Since its publication, the evolution of this domain has been enormous. This book is a unique collection of contributions by researchers who have lived through this evolution, testifying about their personal experiences and sketching the evolution of their respective subdomains since the early years.

When Gottfried Wilhelm Leibniz first arrived in Paris in 1672 he was a well-educated, sophisticated young diplomat who had yet to show any real sign of his latent mathematical abilities. Over his next four crowded, formative years, which Professor Hofmann analyses in detail, he grew to be one of the outstanding mathematicians of the age and to found the modern differential calculus. In Paris, Leibniz rapidly absorbed the advanced exact science of the day. During a short visit to London in 1673 he made a fruitful contact with Henry Oldenburg, the secretary of the Royal Society, who provided him with a wide miscellany of information regarding current British scientific activities. Returning to Paris, Leibniz achieved his own first creative discoveries, developing a method of integral transmutation' through which lie derived the 'arithmetical' quadrature of the circle by an infinite series. He also explored the theory of algebraic equations. Later, by codifying existing tangent and quadrature methods and expressing their algorithmic structure in a universal' notation, lie laid the foundation of formal 'Leibnizian' calculus.

This book is a history, analysis, and criticism of what the author calls "postmodern interpretations of science" (PIS) and the closely related "sociology of scientific knowledge" (SSK). This movement traces its origin to Thomas Kuhn's revolutionary work, The Structure of Scientific Revolutions (1962), but is more extreme. It believes that science is a "social construction", having little to do with nature, and is determined by contextual forces such as the race, class, gender of the scientist, laboratory politics, or the needs of the military industrial complex.Since the 1970s, PIS has become fashionable in the humanities, social sciences, and ethnic or women's studies, as well as in the new academic discipline of Science, Technology, and Society (STS). It has been attacked by numerous authors and the resulting conflicts led to the so-called Science Wars of the 1990s. While the present book is also critical of PIS, it focuses on its intellectual and political origins and tries to understand why it became influential in the 1970s. The book is both an intellectual and a political history. It examines the thoughts of Karl Popper, Karl Mannheim, Ludwik Fleck, Thomas Kuhn, Paul Feyerabend, David Bloor, Steve Woolgar, Steve Shapin, Bruno Latour, and PIS-like doctrines in mathematics. It also describes various philosophical contributions to PIS ranging from the Greek sophists to 20th century post-structuralists and argues that the disturbed political atmosphere of the Vietnam War era was critical to the rise of PIS.

The main part of the third volume of Dr Whiteside's annotated and critical edition of all the known mathematical papers of Isaac Newton reproduces, from the original autograph, Newton's elaborate tract on infinite series and fluxions (the so-called Methodus Fluxionum), including a formerly unpublished appendix on geometrical fluxions. Ancillary documents include, in Part 1, papers on the integration of algebraic functions and, in Part 2, short texts dealing with geometry and simple harmonic motion in a cycloidal arc. Part 3 reproduces, from both manuscript versions of Newton's Lectiones Opticae and from his Waste Book, mathematical excerpts from his researches into light and the theory of lenses at this period. An appendix summarizes mathematical highlights in his contemporary correspondence.

The fifth volume of this definitive edition centres around Newton's Lucasian lectures on algebra, purportedly delivered during 1673-83, and subsequently prepared for publication under the title Arithmetica Universalis many years later. Dr Whiteside first reproduces the text of the lectures deposited by Newton in the Cambridge University Library about 1684. In these much reworked, not quite finished, professional lectiones, Newton builds upon his earlier studies of the fundamentals of algebra and its application to the theory and construction of equations, developing new techniques for the factorizing of algebraic quantities and the delimitation of bounds to the number and location of roots, with a wealth of worked arithmetical, geometrical, mechanical and astronomical problems. An historical introduction traces what is known of the background to the parent manuscript and assesses the subsequent impact of the edition prepared by Whiston about 1705 and the revised version published by Newton himself in 1722. A number of minor worksheets, preliminary drafts and later augmentations buttress this primary text, throwing light upon its development and the essential untrustworthiness of its imposed marginal chronology.

The second volume of Dr Whiteside's annotated edition of all the known mathematical papers of Isaac Newton covers the period 1667-70. It is divided into three parts: Part 1 contains the first drafts of an attempted classification of cubics, together with more general studies on the properties of higher algebraic curves and researches into the 'organic' construction of curves. Part 2 comprises papers on miscellaneous researches in calculus, including the important De Analysi which introduced Newton to John Collins and others outside Cambridge; Newton's original text is here accompanied by Leibniz's excerpts and review, and by Newton's counter review. Part 3 contains Mercator's Latin translation of Kinckhuysen's introduction to algebra, with Newton's corrections and 'observations' upon it, and an account of researches into algebraic equations and their geometrical construction.

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on the localized solution (and grounded in the polemical practices of early Greek science) we see a transition to a practice of mathematics based on the systematic approach (and grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics.

Marking 94 years since its first appearance, this book provides an annotated translation of Sainte-Lague's seminal monograph Les reseaux (ou graphes), drawing attention to its fundamental principles and ideas. Sainte-Lague's 1926 monograph appeared only in French, but in the 1990s H. Gropp published a number of English papers describing several aspects of the book. He expressed his hope that an English translation might sometime be available to the mathematics community. In the 10 years following the appearance of Les reseaux (ou graphes), the development of graph theory continued, culminating in the publication of the first full book on the theory of finite and infinite graphs in 1936 by Denes Koenig. This remained the only well-known text until Claude Berge's 1958 book on the theory and applications of graphs. By 1960, graph theory had emerged as a significant mathematical discipline of its own. This book will be of interest to graph theorists and mathematical historians.

When Newton left Cambridge in April 1696 to take up, at the age of 53, a new career at the London Mint, he did not entirely 'leave off Mathematicks' as he so often publicly declared. This last volume of his mathematical papers presents the extant record of the investigations which for one reason and another he pursued during the last quarter of his life. In January 1697 Newton was tempted to respond to two challenges issued by Johann Bernoulli to the international community of mathematicians, one the celebrated problem of identifying the brachistochrone; both he resolved within the space of an evening, producing an elegant construction of the cycloid which he identified to be the curve of fall in least time. In the autumn of 1703, the appearance of work on 'inverse fluxions' by George Cheyne similarly provoked him to prepare his own ten-year-old treatise De Quadratura Curvarum for publication, and more importantly to write a long introduction to it where he set down what became his best-known statement of the nature and purpose of his fluxional calculus.

This volume reproduces mathematically significant extracts from the extant manuscript record of Newton's researches during 1684-5 into the dynamical motion of bodies under the deviating action of a central force, and his subsequent struggles thereby to explain the observed motions of solar comets and of the moon. The short tract De motu Corporum, which Newton initially composed on this topic in the early autumn of 1684, was primarily built around his earlier proof that in the absence of external perturbation a planetary eclipse may be traversed under an inverse-square force pull to its solar focus, but also discussed the simplest case of resisted ballistic motion. In epilogue, excerpts from his abandoned grand scheme for revising the Principia in the early 1690s detail Newton's planned refinements to his printed exposition of central force, both simplifying and extending it, introducing therein a novel general fluxional measure of such force - but failing adequately to apply it to the primary case of conic motion.

The bringing together, in an annotated and critical edition, of all the known mathematical papers of Isaac Newton marks a step forward in the publication of the works of this great natural philosopher. In all, there are eight volumes in this present edition. Translations of papers in Latin face the original text and notes are printed on the page-openings to which they refer, so far as possible. Each volume contains a short index of names only and an analytical table of contents; a comprehensive index to the complete work is included in Volume VIII. Volume I covers three exceptionally productive years: Newton's final year as an undergraduate at Trinity College, Cambridge, and the two following years, part of which were spent at his home in Lincolnshire on account of the closure of the university during an outbreak of bubonic plague.

This volume reproduces the texts of a number of important, yet relatively minor papers, many written during a period of Newton's life (1677-84) which has been regarded as mathematically barren except for his Lucasian lectures on algebra (which appear in Volume V). Part 1 concerns itself with his growing mastery of interpolation by finite differences, culminating in his rule for divided differences. Part 2 deals with his contemporary advances in the pure and analytical geometry of curves. Part 3 contains the extant text of two intended treatises on fluxions and infinite series: the Geometria Curvilinea (c. 1680), and his Matheseos Universalis Specimina (1684). A general introduction summarizes the sparse details of Newton's personal life during the period, one - from 1677 onwards - of almost total isolation from his contemporaries. A concluding appendix surveys highlights in his mathematical correspondence during 1674-6 with Collins, Dary, John Smith and above all Leibniz.

This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. An important first chapter looks at the mathematicians of the period and how mathematics was perceived by people at large. The central chapters of the book analyse sections of the Collection, identifying features typical of Pappus's mathematical practice. The final chapter draws together the various threads and presents a fuller description of Pappus's mathematical 'agenda'. This is one of few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus's text as part of a wider context and relates it to other contemporary cultural practices and opens avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity.

In this fascinating study, architect and Egyptologist Corinna Rossi analyses the relationship between mathematics and architecture in ancient Egypt by exploring the use of numbers and geometrical figures in ancient architectural projects and buildings. While previous architectural studies have searched for abstract 'universal rules' to explain the history of Egyptian architecture, Rossi attempts to reconcile the different approaches of archaeologists, architects and historians of mathematics into a single coherent picture. Using a study of a specific group of monuments, the pyramids, and placing them in the context of their cultural and historical background, Rossi argues that theory and practice of construction must be considered as a continuum, not as two separated fields, in order to allow the original planning process of a building to re-emerge. Highly illustrated with plans, diagrams and figures, this book is essential reading for all scholars of Ancient Egypt and the architecture of ancient cultures.

Dyma gyfrol sy’n croniclo bywyd a gwaith Griffith Davies (1788–1855), o’i blentyndod tlawd yn ardal chwareli Arfon i’w waith fel sefydlydd ysgolion mathemateg yn Llundain cyn gosod sylfaen i’r proffesiwn actiwari. Tri mis yn unig o addysg ffurfiol a dderbyniodd Davies, a dyna pryd y sylweddolwyd fod ganddo allu rhyfeddol mewn mathemateg. Mentrodd i Lundain, ac ar ôl blynyddoedd wedi ymroi i hunan ddysgu cyhoeddodd lyfrau mewn mathemateg, ac yn y pen draw fe’i penodwyd yn brif actiwari i gwmni yswiriant Guardian yn y ddinas. Wrth i’w yrfa ddatblygu, daeth yn Gymrawd o’r Gymdeithas Frenhinol a derbyniodd glod ac anrhydeddau am ei waith. Bu’n weithgar ym mywyd Cymraeg Llundain gan sefydlu cyfres o ddarlithoedd gwyddonol yn ei famiaith, ac ymgyrchodd yn llwyddiannus dros hawliau tyddynwyr bro ei febyd a thros addysg i’w gyd-wladwyr. Mae hanes bywyd Griffith Davies yn stori sy’n ysbrydoli.

This is a study and translation of the Zhou bi suan jing, a Chinese work on astronomy and mathematics which reached its final form around the first century AD. The author provides the first easily accessible introduction to the developing mathematical and observational practices of ancient Chinese astronomers and shows how the generation and validation of knowledge about the heavens in Han dynasty China related closely to developments in statecraft and politics. The book will be of equal interest to historians of science and those studying the history of Chinese culture.